Rational approximations for $\pi$ using Fibonacci numbers?

Question

It is (well?) known that $$\frac{\pi} 4 = \sum_{k=1}^\infty \arctan \left ( \frac1{F_{2k+1}} \right )$$ Where $F_k$ denotes the $k$-th Fibonacci number. However, any truncation of this sum is irrational!

I am wondering what good rational approximations for $\pi$ are known which use the Fibonacci numbers.

Motivation: I can't find any! Not even on the Wikipedia page.

I am not asking for trivial such approximations, meaning nothing that simply abuses the fact that $F_1 = F_2 = 1$ or $a/a =1$. For example, though $355/113$ is a known good approximation for $\pi$, I don't think $355F_1 / 133$ is any more interesting, nor is $5*355 / (F_5*113)$.

I am not asking for approximations which are obtained using the fact that any integer may be written as a sum of Fibonacci numbers.

I think the following is neat, though not extremely accurate: $$\frac{F_{15}F_{16}+1}{120F_{17}} = 3.1416\dots$$ I found this by accident earlier today, which is what made me wonder what other rational approximations for $\pi$ were known which used the Fibonacci numbers. To keep this question focused, I also say that I prefer approximations which exclusively use integers/Fibonacci numbers, as opposed to using terms from other rational sequences.

I am particularly interested in rational sequences $\{ x_n \}$ which converge to $ \pi$ whose terms are defined using Fibonacci numbers, though I'm not sure if there are any (known). The fact that $ \frac{F_{n}}{F_{n-1}} \to \varphi$ might be useful.

Answer 1

Aha! Just thought of one. Let $$x_n = \frac{F_{n-1}^3}{F_{n}^3}.$$ Now, let $$X_n = \frac{3327}{250}x_n.$$ $X_n$ gives a fairly accurate approximation of $\pi$, though it does not converge to $\pi$.

Edit 1: $X_{30} = 3.14159264\dots$, off in the eighth decimal place. Not a bad approximation!

Edit 2: I came up with this based on the fact that $\frac{\pi}{\varphi^3}(\frac{1000}{1109})=12$. However, about a second after posting this, and as user TonyK points out, I realized this is not true. However, this still produces quite a good (not arbitrarily precise) approximation for $\pi$. Not sure where $3327/250$ comes from.

Answer 2

Since $$\lim_{n\to \infty } \,\frac{\left(F_{n-1}\right){}^3}{\left(F_n\right){}^3}=\sqrt 5-2$$ your constant $k=\frac{3327}{250}$ is the rationalized value of $\frac{\pi }{\sqrt{5}-2}$.

A sequence of the $k$'s would be $$\left\{\frac{40}{3},\frac{133}{10},\frac{173}{13},\frac{2635}{198},\frac{3327}{250} ,\frac{1104391}{82987},\frac{1360570}{102237}, \frac{1390513}{104487},\frac{4178193}{313961},\frac{38997 577}{2930386},\frac{211700657}{15907774},\frac{298052197}{22396468},\frac{937332 361}{70433751},\frac{56537993857}{4248421528},\frac{7 7159305799}{5797964050},\frac{79971302882}{6009265303},\frac{718804393577}{54012 953976}\right\}$$

Computing $$\Delta_n= \log_{10} \left(\left|k_n\frac{ \left(F_{99}\right){}^3}{\left(F_{100}\right){}^3}-\pi \right|\right)$$ we should have the following results $$\left( \begin{array}{cc} n & \Delta_n \\ 1 & -2.22327 \\ 2 & -2.72387 \\ 3 & -4.13879 \\ 4 & -4.71971 \\ 5 & -8.04467 \\ 6 & -8.62783 \\ 7 & -9.67057 \\ 8 & -10.8348 \\ 9 & -12.6291 \\ 10 & -13.6623 \\ 11 & -14.7505 \end{array} \right)$$ For the last of the $k$'s given above and $n=100$ as used for the previous table, we should have $$\color{red}{3.141592653589793238462}46$$

But, we are looping in circles !